论文标题
自由组的原始指数界限和第二个Chebyshev功能
Primitivity index bounds in free groups, and the second Chebyshev function
论文作者
论文摘要
GUPTA和KAPOVICH在双曲线表面上的“无障碍”闭合曲线的结果激励，引入了有限生成的自由组的原始性和简单索引功能，$ d_ {prim}（g; f_n）$和$ d_ {simp}（simp}（g; f_n）$ y $ 1 \ ne g_ $ ne g _ $ and formion and for for。在本文中，我们研究了序列$ d_ {prim}（a^nb^n; f（a，b））$的行为为$ n \ to \ infty $。回答Kapovich的问题，我们证明了此顺序是不受限制的，对于$ n_i = lcm（1,2，\ dots，i）$，我们有$ | d_ {prim}（a^{n_i} b^{n_i}; f（n_i}; f（a，b）;相比之下，我们表明，对于所有$ n \ ge 2 $，一个都有$ d_ {simp}（a^nb^n; f（a，b））= 2 $。除了拓扑和群体理论论点外，数量理论的考虑，尤其是第二个Chebyshev功能的渐近特性，还可以在证明中起关键作用。
Motivated by results about "untangling" closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{prim}(g;F_N)$ and $d_{simp}(g;F_N)$, where $1\ne g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{prim}(a^nb^n; F(a,b))$ as $n\to\infty$. Answering a question of Kapovich, we prove that this sequence is unbounded and that for $n_i=lcm(1,2,\dots,i)$, we have $|d_{prim}(a^{n_i}b^{n_i}; F(a,b))-\log(n_i)|\le o(\log(n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{simp}(a^nb^n; F(a,b))=2$. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.
